> On 14 Dez., 08:15, Virgil <vir...@ligriv.com> wrote:> > > > Then take all paths that cover as many nodes as possible.> >> > Every path in an infinite tree "covers" infinitely many nodes.> > Yes.> >> > > But they must be defined by more than the nodes, if there shall be> > > more than conutably many.> >> > Nonsense! The countably infinite set of nodes necessarily has> > uncountably many subsets, and uncountably many of those subsets will be> > paths.> > You are wrong.

Is WM is declaring the existence of a set which surjects onto its powerset?

Such a claim requires a good deal more proof than, WM has so far been able to produce, and also a good deal more that anyone expects WM to be able to produce.

> I cover all nodes with countably many paths such that> there remains no subset that could represent a real number can be> distinguished by uncovered nodes.

If your ambiguous "subset" reference refers to those sets of nodes including the root node and such that each node of the set has one and only one child node in that set, there are provably uncountably many of them because they can easily be bijected to the set of all subsets of N which is also provably uncountable.> >> > Known infinite sequences may require definition, but there is nothing> > that requires an infinite sequence to be known.> > Everything that in principle can appear as the diagonal of a Cantor> list must be known in principle. Otherwise it could never get known.

Nonsense. If one were to list all known numbers, which according to WM must be possible, since there are, at most only countably many, then the Cantor diagonal process defines a new, and necessarily previously unknown one, so the original "complete listing" is no longer complete.> > > Every one of the> > uncountably many infinite ordered subsets of the ordered set N is an> > infinite sequence and there are uncountably many of them, most of them> > "unknown".> > "There are infinitely many angels", (Cantor, Letter to Jeiler 1888).> Most angels are unknown. But do they belong to mathematics?

As much so, or even more so, that WM does.> > Numbers that are knowable, are knowable by finite definitions. Others> are neither knowable nor elements of mathematics.

They had better be elements of mathematics if their absence would necessarily cause contradictions to exist in mathematics, which it would.> > > > You can also say to construct a collection of nodes, namely the> > > complete Binary Tree, by countably many paths.> >> > While a set of countably many paths may cover all nodes, it necessarily> > omits more paths than it can include.> > There is no path omitted that can be defined by nodes.

There are more of them than WM can count.

Each path bijects with a subset of N, the level numbers of those nodes in that path which are left children of their parent nodes.

Each such subset of N corresponds to a path different from all other paths.

So that WM must be declaring that the number of subsets of N can be counted.

Lets see him try to prove it.

> Recognize and agree or name the nodes of such a path the existence of> which you are asserting.

A path is equally well defined by determining which way it branches from each non-terminal node, and infinite paths do not have terminal nodes.

Every different subset of N defines a unique path, different from the path defined byf any other subset of N, by the construction explained above.This proves the existence of a different path for each different subset of N.

and disproves WM's claims, at least for all mathematics being done outside Wolkenmuekenheim.

> > > > WM has been in here with this shit for over a decade.> >> > > Two wrong assertions in one simple sentence.> >> > Actually WM HAS been here with his shit for over a decade.> > Neither is it shit, nor have I been in sci.logic or sci.math before> 2005.

You have often cited your own nonsense papers. Were they all written post 2005?

> But you will not recognize your errors.

Why should we recognize our few when you refuse to recognize your many?> >> > > Small wonder that you> > > don't understand that Cantor has already been refuted.> >> > Cantor has not been refuted outside of Wolkenmuekenheim.> > But you will not recognize your errors.

Why should we bother to recognize our few when you refuse to recognize your many?> > >> >> > > I construct the complete infinite Binary Tree by means of countably> > > many paths.> >> > You often have claimed so, but since there are many proofs, quite valid> > everywhere here outside of your Wolkenmuekenheim, that you are wrong,> > your "proofs" do not hold up here!> > But you will not recognize your errors.

It is your own errors that cripple your own arguments, and you do not acknowledge any of them.> > >> > > that you ar won If there were more paths definable by nodes, then you> > > should be able to say which one.

We cannot tell which ones are missing without knowing which ones are not missing, but you coyly refuse to tell which ones are missing.

> > > If not, then you should be able to> > > see that only finite definitions define paths like that of 1/3. Alas,> > > there are only countaby many finite definitions. Is it really that> > > hard to understand?> >> > Where is it written that one must be able to name every member of a set> > in order to count its members?> > Without being able to name the elements, one cannot distinguish them.

It is quite possible to distinguish between, say, a positive number and a negative number, without naming either.

It is also quite possible to dstinguish between, say, a rational number and an irrational number, without naming either.

We do both quite frequently.

So that WM's remark is false.

> But a set is undefined unless all elements are distinct.

But not necessarily distinguishable. The set of real numbers is well enough defined, at least outside of Wolkenmuekenheim, even though most of its members are inaccessible > > Extensionality: Sets containing the same elements are equal. If> elements are not nameable, one cannot prove whether two sets are equal> or not.

But one can still often determine whether they are bijectable.> > > The set of reals is known to have more> > members than can be named> > The set of matheologians is known to have more members than can think.> Correction: to have members that cannot think.

Only in the minds to those who dwell in Wolkenmuekenheim, and the opinion in the minds of its non-inhabitants of its inhabitants is considerably less favorable than that.> > > > I construct the complete infinite Binary Tree by means of countably> > > many paths.> >> > Since it is well known that, at last outside Wolkenmuekenheim, the set> > of all possible such paths trivially bijects with the uncountable set of> > all subsets of N, no countable set of paths can include every path.> >> My set includes every path that can be identified by nodes.

List them and I will find as many paths you missed as you have listed.> > >> > > If there were more paths definable by nodes, then you> > > should be able to say which one.> >> > We can find lots of them that you have missed as soon as you tell us> > which ones you have included> > I include all finite paths. The infinite endings are nonsense and not> definable by nodes. I use them only in order to show you that they are> nonsense by proving that you cannot distinguish them by nodes.

Since I have frequently shown that each subset of N can be used to defined a path different from that similarly defined by any other subset of N, and you have never been able to disprove that demonstration, it is established that in an actually complete infinite binary tree there are as many paths as subsets of N.

Until you have some direct disproof of this bijection, you lose.> >> > > If not, then you should be able to> > > see that only finite definitions define paths like that of 1/3.> >> > But the Cantor diagonal process, which proves any listing that you> > present must be incomplete,is totally independent of any need for or> > reliance on finite definitions.> > Wrong. No Cantor list has ever been completely defined and supplied a> complete diagonal number other than by a finite definition.

NOtA BENE: to prove a set is countable at all, it MUST be listable, because the existence of such a list (as a surjection from N to the set in question) is the proof of countability.

Sets which are unlistable are also, by definition, uncountable.> >> > > Alas,> > > there are only countaby many finite definitions. Is it really that> > > hard to understand?> >> > There are perfectly legitimate ways to prove that there have to be more> > paths, and f functions, and subsets of N, than your limited> > definitionism will cover.> > There are perfectly legitmate ways to prove that there are infinitely> many angels. But all that is not part of mathematics.

The mathematical legitimacy of such arguments about angels, absent a whole bunch of assumptions about angels, is, at best. dubious.

Given such a set of assumptions re angels, or anything else, the mathematics can be entirely valid, but will be, as any mathematician or logician would know, conditional upon those assumptions being true.

We ASSUME a set of naturals, a set of rationals, a set of reals, etc., with their usual properties, and then derive other statements conditional on those assumptions being true.

If WM will not work within the standard assumptions mathematics makes, he should stop pretending to be a mathematician.--